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Squaring Binomials

A binomial is an expression composed of two monomials (hence the prefix "bi") that are connected by either a plus sign or a minus sign.

So, how do we square a binomial? Well, we've got a couple of options:

1. Expand and use FOIL
2. Use the formula $a^2\pm2ab+b^2$

Squaring Binomials Using FOIL

Suppose that we are tasked with squaring the binomial $3x - 4$. In order to do this, we must first enclose that binomial in parentheses and give it the appropriate exponent. We want to square the binomial, which is the same as raising it to the second power, thereby creating the following expression: $\left(3x-4\right)^2$.

Once we've done that, it's time for us to go ahead and expand our new expression.

Right! We can expand $\left(3x-4\right)^2$ into $\left(3x-4\right)\left(3x-4\right)$.

Now, all we need to do to finish squaring our binomial is to use FOIL (i.e. multiply the expression's first, outside, inside, and last terms) and simplify the result.

Right:

You'll probably find FOILing to be the most familiar method for squaring a binomial. However, there also exists a shortcut that we can use instead.

Squaring Binomials by Using the Formula $a^2\pm2ab+b^2$

To understand how to use this shortcut, let's first consider the generic binomial $a \pm b$, in which $a$ and $b$ stand for two arbitrary monomials.

Since all binomials follow the form $a \pm b$, a binomial's square will always follow the form of our generic binomial's square: $a^2\pm2ab+b^2$.

This means that, once we've determined which terms in a particular binomial stand for the $a$ and $b$ terms in its generic form, we can use the expression $a^2\pm2ab+b^2$ as a formula for figuring out that binomial's square.

Note: if the binomial we're squaring contains a plus sign, then its product will solely contain plus signs. However, if the binomial we're squaring contains a minus sign, then its square will use a minus sign between the expression's first two terms.

Let's look at an example in which we put this technique to use.

Once we've figured out which terms in our squared binomial stand for the $a$ and $b$ terms in a binomial's generic form, we can then plug those terms into our $a^2+2ab+b^2$ formula and simplify (note that, since the two terms inside of the binomial we're squaring are connected by a plus sign, we're using the version of our formula that solely contains plus signs):

(Videos) Squaring a Binomial

by mathman1024

This video by mathman1024 shows you how to square a binomial.

$\large{\text{Summary}}$

When you see a binomial squared, $(a+b)^2$, your first thought might be to square each term inside, $a^2+b^2$, similar to what we do with radicals. This is incorrect. Remember that squaring something is a short hand way to represent that thing being multiplied by itself. So, $$(a+b)^2=(a+b)(a+b)$$ We can solve this using the FOIL method and combining like terms to see that the answer is in fact different from just squaring the inside terms. $$\begin{gather} {(\color{#8925AE}{a}}+{\color{green}{b}})(\color{#8925AE}{a}+\color{#29A603}{b})\\ F\ \ \ \ \ \ O\ \ \ \ \ \ I\ \ \ \ \ \ L\\ {\color{#8925AE}{a^2}+\color{#8925AE}{a}\color{#29A603}{b}+\color{#29A603}{b}\color{#8925AE}{a}+\color{#29A603}{b^2}}\\ a^2+2ab+b^2 \end{gather}$$

So, as you see, there's actually an extra term in the middle, $2ab$. We can actually use this as a formula now. Instead of FOILing we can square each term and then double the product of them. Also, when it's subtraction we just subtract the middle term. $$\begin{gather}(a+b)^2=a^2+2ab+b^2\\ (a-b)^2=a^2-2ab+b^2 \end{gather}$$

Let's try this out with an example now: $(x+5)^2$. $$\begin{gather}(x+5)^2\\ (x)^2+2(x)(5)+(5)^2\\ x^2+10x+25\end{gather}$$

Try these next ones on yourself!

1. $(x+7)^2$
2. $(x-6)^2$

Squaring Binomials With Distribution

How do we square binomials? One way is to multiply a binomial by itself using a table, but there are even easier ways! Let's look at an example problem that tasks us with finding the square of the binomial $3x + 7$:

$$(3x+7)^2=?$$

When dealing with problems like these, we can expand the initial expression by using the distributive property:

$$(3x+7)^2=(3x+7)(3x+7)$$ $$=3x(3x+7)+7(3x+7)$$ $$=9x^2+21x+21x+49$$ $$=\boxed{9x^2+42x+49}$$

Now, let's try another example.

Expand $(4x-2y)^2$.

Again, we expand the expression by using distribution:

$$(4x-2y)^2=(4x-2y)(4x-2y)$$ $$=4x(4x-2y)-2y(4x-2y)$$ $$=16x^2-8xy-8xy+4y^2$$ $$=\boxed{16x^2-16xy+4y^2}$$

At this point, you may be wondering whether or not we could be using a formula to square binomials instead. While such a formula does exist, I prefer not to commit it to memory. After a lot of practice, this whole procedure becomes automatic, and a formula is no longer needed.

In order to derive this formula, we'll be expanding the expression $(a+b)^2$, where $a$ and $b$ stand for two arbitrary terms.

On your own, try using the distributive property to come up with the expanded version of $(a+b)^2$.

Squared Binomial Formula

When we expand the expression $(a+b)^2$, we get:

$$(a+b)^2 = (a+b)(a+b)$$ $$=a(a+b)+b(a+b)$$ $$=a^2+ab+ab+b^2$$ $$=\boxed{a^2+2ab+b^2}$$

This is our formula. Now, let's show that this formula works by testing it out on one of our previous example expressions: $(4x-2y)^2$. First, we'll let $a=4x$ (since $4x$ is our binomial's first term), and we'll let $b=-y$ (since $-y$ is our binomial's last term). Then, we plug those terms into our formula and simplify:

$$(4x-2y)^2=(4x)^2+2(4x)(-2y)+(-2y)^2$$ $$=16x^2+(-16xy)+4y^2$$ $$=\boxed{16x^2-16xy+4y^2}$$

Notice that this is the same exact answer that we got when we previously used the distributive property to expand $(4x-2y)^2$.